You can make non-petroleum based plastics, there are non-petroleum based jet fuel, there are even non-petroleum based vehicle fuels. Don't tell me we are so dependent on petroleum that when the world supply runs out we're suddenly back into the stone age.
Sure, we could probably get by without natural crude oil. It would be darned difficult, stupidly expensive, send everyone's cost of living way up, it would take a ridiculous amount of time and effort to switch all of the infrastructure over, and it would generally suck for all and sundry. But yes, we could do it.
The real question is why on earth would we want to? We have plenty of natural crude in the ground that we would be easier to use, cost much less, and work far better. The only reason we don't use it is that we choose not to. Or rather, some of us force the rest of us not to.
So, why on earth shouldn't we use the oil in ANWR?